To simplify the expression [tex]\(\sqrt[4]{162}\)[/tex], follow these detailed steps:
1. Prime Factorization: First, we need to find the prime factorization of 162.
- 162 is divisible by 2: [tex]\(162 \div 2 = 81\)[/tex].
- 81 is divisible by 3: [tex]\(81 \div 3 = 27\)[/tex].
- 27 is divisible by 3: [tex]\(27 \div 3 = 9\)[/tex].
- 9 is divisible by 3: [tex]\(9 \div 3 = 3\)[/tex].
- 3 is divisible by 3: [tex]\(3 \div 3 = 1\)[/tex].
Therefore, the prime factors of 162 can be written as:
[tex]\[
162 = 2 \times 3^4
\][/tex]
2. Express Under the Radical Form: We will now express [tex]\(\sqrt[4]{162}\)[/tex] using the prime factorization:
[tex]\[
\sqrt[4]{162} = \sqrt[4]{2 \times 3^4}
\][/tex]
3. Separate the Factors: Use the property of radicals that [tex]\(\sqrt[4]{a \times b} = \sqrt[4]{a} \times \sqrt[4]{b}\)[/tex]:
[tex]\[
\sqrt[4]{2 \times 3^4} = \sqrt[4]{2} \times \sqrt[4]{3^4}
\][/tex]
4. Simplify Each Factor:
- For [tex]\(\sqrt[4]{3^4}\)[/tex], simplify by noting that the fourth root of [tex]\(3^4\)[/tex] is simply 3:
[tex]\[
\sqrt[4]{3^4} = 3
\][/tex]
- For [tex]\(\sqrt[4]{2}\)[/tex], we cannot simplify it further, so it remains as is.
5. Combine the Results: Putting these simplified parts together gives us:
[tex]\[
\sqrt[4]{2} \times 3
\][/tex]
By convention, we write the integer part first:
[tex]\[
3 \sqrt[4]{2}
\][/tex]
Therefore, the simplified radical form of [tex]\(\sqrt[4]{162}\)[/tex] is:
[tex]\[
3 \sqrt[4]{2}
\][/tex]
